Anyone who has ridden in airplane knows the thrill of taking off from the runway. When the airplane is cleared for takeoff, the pilot releases the brakes and controls the engines to increase thrust. The airplane begins moving down the runway faster and faster. When the airplane is moving fast enough so that the amount of lift generated by the wings permits safe takeoff, the pilot controls the plane's flight control surfaces (e.g., the elevator) to cause the nose of the airplane to rotate skyward. The airplane leaves the ground with its nose pitched upwards. Depending upon the particular aircraft configuration, weight, weather and other factors, the plane gains altitude at different rates as it ascends to a desired altitude for level flight.
Certain aircraft performance speeds known as “v-speeds” change based on operating conditions. One of the “v-speeds” is VR—also known as takeoff rotation speed. VR is the speed at which aircraft takeoff rotation is initiated by the pilot. VR is always greater than another v-speed (VMCA) which is the minimum speed which provides directional control in the air during engine failure. V-speeds can be affected by a variety of factors including gross takeoff weight, air pressure, temperature, aircraft configuration and runway conditions.
Aircraft takeoff distance is classically divided in two segments: ground acceleration (d1) and air transition (d2). See FIG. 1.
Ground acceleration distance d1 is measured from when the brakes are released to the beginning of rotation at VR. Air transition distance d2 is measured from that point to the point d1 where the aircraft reaches a specified height (h) above the runway.
Usually the length of segment d1 is calculated by mathematically integrating movement equations reflecting the engines thrust, airplane lift, drag and weight in given atmospheric conditions.
For segment d2, based on the energy conservation principle, the following equation can be derived:
                                                                                                              1                    2                                    ⁢                  m                  ⁢                                                                          ⁢                                      V                    R                    2                                                                                                      1                                              +                                                                                          ∫                    r                                    ⁢                                                            (                                                                        T                          →                                                -                                                  D                          →                                                                    )                                        ·                                          ⅆ                                              r                        →                                                                                                                                                2                                                    =                                                                                                  1                    2                                    ⁢                                                                          ⁢                  m                  ⁢                                                                          ⁢                                      V                    SH                    2                                                                                                      3                                              +                                                    mgh                                                                    4                                                                        [        1        ]            
where:
Expression 1: kinetic energy of the airplane at beginning of rotation
Expression 2: total energy increase due to the engines thrust along the flight path
Expression 3: kinetic energy of the airplane at the end of air transition
Expression 4: potential gravitational energy at the end of air transition.
For simplification and better understanding, consider the case of a given takeoff from a level runway. In this case, the values of m, g, VR and h are previously known. As a consequence, expressions 1 and 4 are also predetermined. Therefore, the larger expression 3 becomes, the larger expression 2 will also be. In other words, the higher VSH is, the longer the flight path (represented by “r”) is likely to be.
Detailed mathematical modeling of air transition can be difficult due to a number of factors that are not trivial to evaluate, such as variable ground effect, transient aerodynamics and the effect of piloting technique. Therefore, segment d2 is usually calculated by using simplified parametric models entirely based on flight test data.
Classically, the approved length of segment d2 is based on operational procedures that demand the airplane to rotate to a given pitch attitude (θ) at a given speed (VR) and with a given pitch rate (q). In both flight testing and daily operation, these parameters (θ, q and VR,) are subjected to a series of constraints, for example:
Pitch Angle (θ):                Can be limited by the airplane capacity to achieve the minimum gradients of the applicable certification requirements with one engine made inoperative, in any condition within the operational envelope of the aircraft        By geometrical limitations, in order to avoid tail strikes during takeoff        
Pitch Rate (q):                Low pitch rates lead to longer takeoff distances        Excessively high rates introduce the risk of overshooting the defined takeoff θ, and may increase the risk of tail strikes        
Rotation Speed (VR):                VR is constrained by requirements that relate it to the typical takeoff speeds (for Part-23 and Part-25 certifiable airplanes, V1, VLOF and V2)        VR shall be such that, after rotation, the airplane will accelerate to a given speed before reaching a given height (V2 at 35 ft for airplanes certifiable under Part-23 and Part-25 of Title 14 of the regulations of the Federal Aviation Administration, incorporated herein by reference), while achieving the minimum climb gradients required during takeoff.        
In order to comply with all the applicable requirements while not demanding exceptional skill from the pilot, aircraft manufacturers will usually define a fixed takeoff pitch for each takeoff configuration, to be commanded at VR with a given rotation rate.
When performing flight tests for determining the nominal aircraft takeoff distance, manufacturers may follow these procedures, so that the measured performance can be reproduced in actual operation. In addition to the aforementioned limitations, a reasonable amount of data dispersion is introduced into performance models during flight tests due to natural differences in piloting actions, from pilot to pilot. As a result, the aircraft short-field performance can be severely affected by these operational constraints, leading to a non-optimal takeoff distance calculation model.
Another aspect of current practices is that the takeoff attitude is generally defined to satisfy the climb gradient requirements for the most unfavorable condition within the aircraft operation and loading envelopes.
The climb gradient γ is defined as follows:
                    γ        =                                            T              -              D                        W                    -          ϕ                                    [        2        ]            
In the equation above, T is the net engines thrust, D is the airplane drag in the specified configuration, W is the airplane weight and φ is the runway slope. Conditions for calculation of T and D are defined in the applicable Part-23 and Part-25 certification requirements mentioned above, and incorporated herein by reference.
However, most times (and especially in short airfields) the actual condition is such that, with the defined pitch, the actual climb gradients achievable are much higher than required (there is a so called “energy excess”), resulting in great acceleration at expense of a shallow takeoff flight path. This leads to time increase from rotation to the end of air segment, and consequently produces a longer d2. The following examples illustrate these effects.